## Geometric Abelian Class Field Theory

##### Summary

The purpose of this thesis has been twofold. First to give a detailed treatment of
unramified geometric abelian class field theory concentrating on Deligne's geometric
proof in order to remedy the unfortunate situation that the literature on this topic is
very deficient, partial and sketchy written(1). In the second place to give also a detailed
treatment of ramified geometric abelian class field theory and more importantly to
find a new geometric proof for the ramified theory by trying to adapt or to lean on
Deligne's geometric argument in the unramified case.
What was achieved is the following: we begin with discussing and building up the
unramified theory in details, which describes a remarkable connection between the
Picard group and the abelianized etale fundamental group of a smooth, projective,
geometrically irreducible curve over a finite field. We give the necessary background
culminating in the fully presented geometric proof of Deligne. Then we turn to
a detailed discussion of the tamely ramified theory, which transforms the classical
situation to the open complement of a finite set of points of the curve, establishing
a connection between a modified Picard group and the tame fundamental group of
the curve with respect to this finite subset of points. In the end we finally present a
geometric proof for the tamely ramified theory.
(1) Probably due to the fact that people working on this ?eld are mostly concentrating on the higher
dimensional theory, that is on the geometric Langlands program, instead of writing down relatively
classical works in a fairly didactic way.